ON A PROBLEM OF SCHOENBERG AND WILLS IN DIOPHANTINE APPROXIMATION

Abstract

Let n be a fixed natural number. Wills has shown that there exist irrational numbers α1,..,αnand real numbers β1 with max1≤i≤n ∥q7alpha;i−βi>1/2 – 1/2n for all integers q (‖·‖ denotes the distance to the nearest integer). His example is αi=αand βi=i/n +δ, δ suitably chosen. Beyond that, he asked if αican be found with pairwise different ‖αi|. We prove that this does not hold for n ≤ 5, thereby revealing the close relation to Schoenberg's billiard ball problem for cubes and classifying its critical lines in these dimensions.